Localized free-flow bounds and compactness mechanism

Put $M={\sup }_j\mathrm{\parallel }{y}_{j,0}{\mathrm{\parallel }}_{H_D^{\tau }}$. By the scale inclusion, ${\sup }_j\mathrm{\parallel }{y}_{j,0}{\mathrm{\parallel }}_2\le CM$, and weak convergence in $H_D^{\tau }$ implies weak convergence in $L^2$: compose any bounded $L^2$ functional with the continuous inclusion. Unitarity gives ${\sup }_{j,t}\mathrm{\parallel }{y}_j(t){\mathrm{\parallel }}_2\le CM$. The smoothing estimate followed by the fractional identification and restriction gives ${\sup }_j\mathrm{\parallel }\psi y_j{\mathrm{\parallel }}_{L^2(0,T;H^{1\mathrm{/}2}(G))}\le CM$. The cutoff multiplication and restriction are bounded on the relevant spaces, so the resulting time-valued classes are strongly measurable.

For $z\in L^2(\mathrm{\Omega })$ define $D_{\psi }z\in (H_0^2(G){)}^{\mathrm{\prime }}$ first on $\zeta \in C_c^{\mathrm{\infty }}(G)$ by $(D_{\psi }z)(\zeta )=i⟨z,\mathrm{\Delta }(\psi \zeta ){⟩}_{L^2(\mathrm{\Omega })}$, extending $\psi \zeta$ by zero outside $G$. The product rule, bounded derivatives of $\psi$ and Cauchy–Schwarz give $\mathrm{\mid }(D_{\psi }z)(\zeta )\mathrm{\mid }\le C\mathrm{\parallel }z{\mathrm{\parallel }}_2\mathrm{\parallel }\zeta {\mathrm{\parallel }}_{H^2(G)}$. Density in the definition of $H_0^2(G)$ extends this to a bounded linear map $D_{\psi }:L^2(\mathrm{\Omega })\to (H_0^2(G){)}^{\mathrm{\prime }}$. In particular $t\mapsto D_{\psi }{y}_j(t)$ is norm-continuous, hence strongly measurable, and its norm is uniformly bounded by $CM$.

Let $g\in L^2(0,T;L^2(\mathrm{\Omega }))$. Strong continuity of $W(-t)$, its unitary bound, and approximation of $g$ by simple functions make $t\mapsto W(-t)(\psi g(t))$ strongly measurable. Its norm has a finite integral by Cauchy–Schwarz on $(0,T)$, so $h={\int }_0^{T}W(-t)(\psi g(t))dt$ exists in $L^2(\mathrm{\Omega })$. Because $\psi$ is real and $W(t{)}^{\ast }=W(-t)$, moving the bounded pairing through this Bochner integral yields ${\int }_0^T⟨\psi y_j(t),g(t)⟩dt=⟨y_{j,0},h⟩$. The right side tends to zero by step 1.1.

Put $Y=(H_0^2(G){)}^{\mathrm{\prime }}$ and let $J:L^2(G)\to Y$ be the bounded linear pivot $Jv(\zeta )=⟨v,\zeta ⟩$ from [F3]. For $u\in L^2(\mathrm{\Omega })$ write $w_u(t)=J(\psi W(t)u{\mathrm{\mid }}_G)$. The maps $w_u$ and $D_{\psi }W(t)u$ are norm-continuous and uniformly bounded in $Y$, by boundedness of $J$, cutoff restriction, $D_{\psi }$ and unitarity. Their products with a scalar time test or its derivative are therefore Bochner integrable by [F5]. For $u\in D({\mathrm{\Delta }}_D)$, self-adjointness and the evolution equation give ${\mathrm{\partial }}_t⟨\psi W(t)u,\zeta ⟩=i⟨W(t)u,\mathrm{\Delta }(\psi \zeta )⟩$ for every $\zeta \in C_c^{\mathrm{\infty }}(G)$; the zero extension of $\psi \zeta$ is in $C_c^{\mathrm{\infty }}(\mathrm{\Omega })\subset D({\mathrm{\Delta }}_D)$. Fix $\phi \in C_c^{\mathrm{\infty }}((0,T);\mathbb{C})$. Scalar integration by parts yields the time-integrated identity after evaluation on every such $\zeta$. Evaluation $F\mapsto F(\zeta )$ is a bounded complex-linear map on $Y$, so [F5] commutes it with both Bochner integrals. Density of $C_c^{\mathrm{\infty }}(G)$ in $H_0^2(G)$ then makes their difference the zero continuous functional, giving ${\int }_0^T{w}_u(t){\phi }^{\mathrm{\prime }}(t)dt=-{\int }_0^T{D}_{\psi }W(t)u\phi (t)dt$ in $Y$. For general $u\in L^2(\mathrm{\Omega })$ choose $u_k\in D({\mathrm{\Delta }}_D)$ with $u_k\to u$ in $L^2$; unitarity and the bounded maps just used give uniform-in-time convergence of both $Y$-valued factors, hence convergence of their integrals against ${\phi }^{\mathrm{\prime }}$ and $\phi$. The same identity follows for $u$. This is precisely the derivative identity in [F4], with no conjugation of $\phi$; thus ${\mathrm{\partial }}_t(\psi W(t)u{\mathrm{\mid }}_G)=D_{\psi }W(t)u$ after the pivot identification. Step 2.1 now bounds the derivative family in $L^{\mathrm{\infty }}(0,T;H^{-2}(G))$, and integration of that uniform norm bound gives the $L^1(0,T;H^{-2}(G))$ bound multiplied by $T$. This argument uses the conjugate-dual pairing; it does not assert a boundary trace of $y_j$.

The bounded smooth set $G$ and the triple in [F3], together with steps 1.1 and 3.1, satisfy the hypotheses of the compactness input [F4]. Hence $(\psi y_j{\mathrm{\mid }}_G)$ is relatively compact in $L^2(0,T;L^2(G))$. Since $\psi$ vanishes outside $G$ as a function on $\mathrm{\Omega }$, zero extension from $G$ is an $L^2$ isometry on these functions. Their zero extensions are therefore relatively compact in $L^2(0,T;L^2(\mathrm{\Omega }))$.

Every subsequence has a further subsequence converging strongly in $L^2(0,T;L^2(\mathrm{\Omega }))$ by step 4.1. If its limit is $v$, take $g=v$ in step 2.2 and pass to the strong limit; Cauchy–Schwarz gives $\mathrm{\parallel }v{\mathrm{\parallel }}_{L_t^2{L}_x^2}^2=0$. Thus every such limit is zero. If the original sequence did not converge strongly to zero, a subsequence would stay a fixed positive distance from zero, contradicting the preceding conclusion for its strongly convergent further subsequence. This proves the assertion.